Prime-Power Rarefaction and a Density-One Lower Bound for Erd\H{o}s Problem 400

Abstract

For fixed $k\ge 2$, let $g_k(n)$ be the greatest excess $a_1+\cdots+a_k-n$ among positive integers $a_i$ satisfying $a_1!\cdots a_k!\mid n!$. We prove that, for every $\varepsilon>0$, all but $o(x)$ integers $n\le x$ satisfy \[ g_k(n)\ge \left(\frac{3(k-1)}{\log 12}-\varepsilon\right)\log n. \] We also prove, as $n\to\infty$, the pointwise upper bound \[ g_k(n)\le (k-1)\log_2 n+\log_2\log n+O_k(1). \] The central analytic input is uniform phase separation for one or two frequencies on fixed-prime $S$-unit progressions, deduced directly from the finite exceptional-subspace alternative of Drmota and Spiegelhofer, and the resulting uniform digit-sum normal-order theorem. A mixed $2$--$3$ representation, quantitative two-block estimates, and a large-prime Kummer sieve produce the stated coefficient.